I wrote previously that the Black-Scholes options pricing model replaces the equity drift μ with r - ½σ², where r is the riskless interest rate and σ is the stochastic element of the equity's price movement. This substitution has the effect of eliminating the equity drift coefficient from the model, and obviating the need to decide what the drift coefficient actually is.
The reasoning given by Black and Scholes in their paper that introduced their model (as well as one of the very first uses of risk-neutral pricing generally) were (1) that, through continuous rebalancing, the second-order (dz²) stochastic terms in the return on a portfolio became riskless (this argument implicitly relies upon Itō's Lemma, although the paper did not explicitly refer to it), and (2) that even if they were not riskless due to discontinuity, the second-order terms should be subject to the riskless interest rate because they are uncorrelated with the market rate of return. Thus, they argued, second-order terms must be priced with the riskless interest rate directly, and any risk premium must apply only to any first-order stochastic terms.
This can only be achieved by adapting only the drift coefficient -- and not the stochastic coefficient -- in constructing a risk-neutral probability distribution. This means replacing the equity drift μ with r - ½σ².
And this is all roughly consistent with traditional utility theory.
Showing posts with label finance. Show all posts
Showing posts with label finance. Show all posts
Wednesday, December 12, 2012
Wednesday, December 5, 2012
The Lognormal Distribution and Risk-Neutral Pricing
I wrote previously that the ergodic property of the normal distribution is so useful that it often makes sense to assume a normal distribution even in the face of evidence to the contrary.
Before going to that extreme, however, it is sometimes possible to arrive at a normal distribution by looking at a function of an original variable rather than the variable itself. One example of this is the lognormal distribution, which is a distribution whose log is a normal distribution. In finance, the future price of a stock is often considered to have a lognormal distribution, which gives the rate of return on the stock a normal distribution.
In considering a lognormal distribution, we generally refer to aspects of its log. In particular, we usually define a particular lognormal distribution based upon the mean, μ, and variance, σ², of the log.
There are any number of things to be known about lognormal distributions, but the most important fact about them for my purposes is that the expected value of a lognormal distribution is exp(μ + ½σ²).
Getting back to finance, the future price of a stock at time t can be considered to have a lognormal distribution with log-mean lnS + μt and log-variance σ²t, where S is the price of the stock at time 0. (here I have effectively used μ and σ² as the mean and variance of the instantaneous rate of return of the stock, so that at any given point in time in the future t the rate of return on the stock will have mean μt and variance σ²t). This gives the expected value of the stock at time t as exp(lnS + μt + ½σ²t).
Under a risk-neutral pricing regime, however, the expected future price of the stock should be exp(lnS + rt), where r is the riskless interest rate. So if we want to preserve the lognormal distribution of the stock price, we somehow have to adapt our real world expectations into risk-neutral probabilities so that exp(lnS + μt + ½σ²t) equals exp(lnS + rt), or more simply so that μ + ½σ² equals r.
One method to accomplish this is simply to replace μ with r - ½σ². And this is precisely what the Black-Scholes options pricing model does, about which more later.
Before going to that extreme, however, it is sometimes possible to arrive at a normal distribution by looking at a function of an original variable rather than the variable itself. One example of this is the lognormal distribution, which is a distribution whose log is a normal distribution. In finance, the future price of a stock is often considered to have a lognormal distribution, which gives the rate of return on the stock a normal distribution.
In considering a lognormal distribution, we generally refer to aspects of its log. In particular, we usually define a particular lognormal distribution based upon the mean, μ, and variance, σ², of the log.
There are any number of things to be known about lognormal distributions, but the most important fact about them for my purposes is that the expected value of a lognormal distribution is exp(μ + ½σ²).
Getting back to finance, the future price of a stock at time t can be considered to have a lognormal distribution with log-mean lnS + μt and log-variance σ²t, where S is the price of the stock at time 0. (here I have effectively used μ and σ² as the mean and variance of the instantaneous rate of return of the stock, so that at any given point in time in the future t the rate of return on the stock will have mean μt and variance σ²t). This gives the expected value of the stock at time t as exp(lnS + μt + ½σ²t).
Under a risk-neutral pricing regime, however, the expected future price of the stock should be exp(lnS + rt), where r is the riskless interest rate. So if we want to preserve the lognormal distribution of the stock price, we somehow have to adapt our real world expectations into risk-neutral probabilities so that exp(lnS + μt + ½σ²t) equals exp(lnS + rt), or more simply so that μ + ½σ² equals r.
One method to accomplish this is simply to replace μ with r - ½σ². And this is precisely what the Black-Scholes options pricing model does, about which more later.
Sunday, December 2, 2012
Laws of Price: Prices Are Set At The Margin
In a free market, a prevailing price means just one thing: that the demand and the supply are (for a moment) equal at that price.
In a grand sense, this is not a very powerful meaning: we conventionally think of the value of portfolios as the sum of the products of the positions in every investment and the last price of the particular investment. So with 1,000,000 shares of IBM trading at $200.00 a share, this would have a value of $200,000,000.00. But if someone holding this position were to try to convert it to cash all at once, he would almost certainly net less than $200,000,000.00: his position is equivalent to about 25% of the average daily volume of the stock, and it is unlikely that buyers would arrive conveniently to absorb the additional sales that he has introduced, and leave the stock price unaffected.
But in a microscopic sense, it is quite a powerful meaning. In particular, if there is reason to believe that a realistic investor would want to buy rather than sell (or vice versa) a particular position at a given price, then it does not matter that this tendency would be exhausted with just a tiny position: the price cannot be expected to hold, other things being equal.
The latter fact is central to my understanding of the pricing of financial risk.
In a grand sense, this is not a very powerful meaning: we conventionally think of the value of portfolios as the sum of the products of the positions in every investment and the last price of the particular investment. So with 1,000,000 shares of IBM trading at $200.00 a share, this would have a value of $200,000,000.00. But if someone holding this position were to try to convert it to cash all at once, he would almost certainly net less than $200,000,000.00: his position is equivalent to about 25% of the average daily volume of the stock, and it is unlikely that buyers would arrive conveniently to absorb the additional sales that he has introduced, and leave the stock price unaffected.
But in a microscopic sense, it is quite a powerful meaning. In particular, if there is reason to believe that a realistic investor would want to buy rather than sell (or vice versa) a particular position at a given price, then it does not matter that this tendency would be exhausted with just a tiny position: the price cannot be expected to hold, other things being equal.
The latter fact is central to my understanding of the pricing of financial risk.
Sunday, October 28, 2012
Risk-Neutral Pricing
I have previously introduced the concept of the "risk premium" in investing, which holds that "risky" investments should have a higher expected rate of return than "non-risky" investments. In that earlier entry, I noted that Utility Theory provides a framework for explaining why the risk premium should exist. In this entry, I will try to explain a useful way of quantifying the disposition of risk premium: "risk-neutral pricing."
Typically, the risk premium will be spoken of in terms of "expected rate of return," as in, "the expected rate of return on common stocks is around 6%, while the expected rate of return on cash is around 1%." The higher expected rate of return stated here for stocks as opposed to cash indicates a higher risk premium for stocks.
However, when option securities are considered, this way of thinking runs aground. We might consider a call option, for example, that pays in dollars the amount (if any) by which a particular stock index exceeds a given value at close of trading on a particular day, or nothing if the stock index is less than or equal to the value. But now how do we come up with an expected rate of return for this option?
If we look at the Put-Call Parity equation,
C - P = S - X * exp(-rT),
we see that on the right hand side of the equation are terms, S and X * exp(-rT), that should have the expected returns of stocks and of cash, respectively (cash being a risk-free asset, subject to the risk-free interest rate of r).
So what if we treat the call as the difference between the index value -- discounted at the expected returns on stocks -- and a riskless payment of the strike price -- discounted at the risk-free interest rate -- on the expiration date?
This method falls apart pretty quickly. Suppose we look at just a piece of the call using this perspective: the portion from where the index equals the strike price to where it's the strike price plus one. Let's suppose also that the strike price is 1000 and the expiration date is one year in the future. So the value of the stock piece, using the expected rates of return above, will be between 1000 and 1001 discounted by 6%, or between about 943.40 and 944.34. But the value of the risk-free piece will be 1000 discounted by 1%, or about 990.10. This suggests that the value of this portion of the call option will be between -46.70 and -45.76: negative values for something that can only ever result in positive cash flow! No rational person will give you money to accept the possibility that he might give you more money later on.
The way around this is to discount everything at the risk-free interest rate: both the stock portion and the strike price portion. This gives the value of the portion of the call option we're looking at as being between 0 and 0.99, which is very consistent with the value at the end of the year being between 0 and 1.
But where does the risk premium appear?
The risk premium appears in adjustments to the probabilities of different outcomes. In particular, prices will be determined using probabilities that are different from actual expectations.
In looking at the portion of the call option where the final index value is between the strike price and the strike price plus one, I never considered the probability of this outcome: I essentially took each potential outcome -- from the index being equal to the strike price up to being that value plus one -- as given, and looked at what the value of the option would be under that outcome. To get an actual value of the call option, I'd have to assign a probability to each of the outcomes, and generate a weighted average for the option's value under all possible outcomes. In essence, the risk premium on stocks is created because the pricing probability assumes that high returns on stocks are less likely than actual expectations.
So, why use the risk-less interest rate rather than some other rate?
First of all, it would probably be more accurate to use actual expectations, and then have a different expected rate of return apply to each possible outcome. But this would create a mess: how do you figure out what expected rate of return to apply to a given outcome, and from where (without a LOT of work) do you get actual expectations?
Risk-neutral pricing takes advantage of the fact that prices based on uncertain events are based on the product of the probability of given outcomes and the expected rate of return that applies to that outcome. So rather than (more accurately) using actual expectations for the probabilities and some yet-to-be-determined expected rate of return for each outcome, risk-neutral pricing uses a single expected rate of return (the risk-free interest rate) and adjusted probabilities to come up with the same price that would result from the theoretically more accurate method of using actual expectations and variable expected rates of return.
So, again, why use the risk-less interest rate rather than some other rate?
The class of securities that by definition earn the risk-free interest rate -- riskless securities -- place some constraints on the adjusted pricing probabilities: the price indicated, under whatever pricing algorithm we use, for a fixed amount paid under all possible outcomes must be consistent with the risk-free interest rate. If we try to do this using some other interest rate, we will have to have our adjusted probabilities sum to less than 1 if we use an interest rate lower than the risk-free interest rate, and we will have to have our adjusted probabilities sum to more than 1 if we use an interest rate higher than the risk-free rate. The adjusted probabilities would not be proper probability distribution functions, because a proper probability distribution function will sum to exactly 1. And, if a function that did not sum to 1 was used, it would create immediate arbitrage potential in either selling a payoff under all outcomes if the function sums to less than 1, or buying a payoff under all outcomes if the function sums to more than 1
So, it is ultimately simplicity that drives us to use a single interest rate and adjusted probabilities rather than variable expected rates of returns and actual expectations. Then, having limited ourselves in this way out of concern for simplicity, we are limited to using the risk-free interest rate as our single expected rate of return for the reason given above.
Finally, the reason this method is called "risk-neutral pricing" is that, using the adjusted pricing probabilities, all securities are assumed to have an equal expected rate of return, which is consistent with a risk-neutral market.
Typically, the risk premium will be spoken of in terms of "expected rate of return," as in, "the expected rate of return on common stocks is around 6%, while the expected rate of return on cash is around 1%." The higher expected rate of return stated here for stocks as opposed to cash indicates a higher risk premium for stocks.
However, when option securities are considered, this way of thinking runs aground. We might consider a call option, for example, that pays in dollars the amount (if any) by which a particular stock index exceeds a given value at close of trading on a particular day, or nothing if the stock index is less than or equal to the value. But now how do we come up with an expected rate of return for this option?
If we look at the Put-Call Parity equation,
C - P = S - X * exp(-rT),
we see that on the right hand side of the equation are terms, S and X * exp(-rT), that should have the expected returns of stocks and of cash, respectively (cash being a risk-free asset, subject to the risk-free interest rate of r).
So what if we treat the call as the difference between the index value -- discounted at the expected returns on stocks -- and a riskless payment of the strike price -- discounted at the risk-free interest rate -- on the expiration date?
This method falls apart pretty quickly. Suppose we look at just a piece of the call using this perspective: the portion from where the index equals the strike price to where it's the strike price plus one. Let's suppose also that the strike price is 1000 and the expiration date is one year in the future. So the value of the stock piece, using the expected rates of return above, will be between 1000 and 1001 discounted by 6%, or between about 943.40 and 944.34. But the value of the risk-free piece will be 1000 discounted by 1%, or about 990.10. This suggests that the value of this portion of the call option will be between -46.70 and -45.76: negative values for something that can only ever result in positive cash flow! No rational person will give you money to accept the possibility that he might give you more money later on.
The way around this is to discount everything at the risk-free interest rate: both the stock portion and the strike price portion. This gives the value of the portion of the call option we're looking at as being between 0 and 0.99, which is very consistent with the value at the end of the year being between 0 and 1.
But where does the risk premium appear?
The risk premium appears in adjustments to the probabilities of different outcomes. In particular, prices will be determined using probabilities that are different from actual expectations.
In looking at the portion of the call option where the final index value is between the strike price and the strike price plus one, I never considered the probability of this outcome: I essentially took each potential outcome -- from the index being equal to the strike price up to being that value plus one -- as given, and looked at what the value of the option would be under that outcome. To get an actual value of the call option, I'd have to assign a probability to each of the outcomes, and generate a weighted average for the option's value under all possible outcomes. In essence, the risk premium on stocks is created because the pricing probability assumes that high returns on stocks are less likely than actual expectations.
So, why use the risk-less interest rate rather than some other rate?
First of all, it would probably be more accurate to use actual expectations, and then have a different expected rate of return apply to each possible outcome. But this would create a mess: how do you figure out what expected rate of return to apply to a given outcome, and from where (without a LOT of work) do you get actual expectations?
Risk-neutral pricing takes advantage of the fact that prices based on uncertain events are based on the product of the probability of given outcomes and the expected rate of return that applies to that outcome. So rather than (more accurately) using actual expectations for the probabilities and some yet-to-be-determined expected rate of return for each outcome, risk-neutral pricing uses a single expected rate of return (the risk-free interest rate) and adjusted probabilities to come up with the same price that would result from the theoretically more accurate method of using actual expectations and variable expected rates of return.
So, again, why use the risk-less interest rate rather than some other rate?
The class of securities that by definition earn the risk-free interest rate -- riskless securities -- place some constraints on the adjusted pricing probabilities: the price indicated, under whatever pricing algorithm we use, for a fixed amount paid under all possible outcomes must be consistent with the risk-free interest rate. If we try to do this using some other interest rate, we will have to have our adjusted probabilities sum to less than 1 if we use an interest rate lower than the risk-free interest rate, and we will have to have our adjusted probabilities sum to more than 1 if we use an interest rate higher than the risk-free rate. The adjusted probabilities would not be proper probability distribution functions, because a proper probability distribution function will sum to exactly 1. And, if a function that did not sum to 1 was used, it would create immediate arbitrage potential in either selling a payoff under all outcomes if the function sums to less than 1, or buying a payoff under all outcomes if the function sums to more than 1
So, it is ultimately simplicity that drives us to use a single interest rate and adjusted probabilities rather than variable expected rates of returns and actual expectations. Then, having limited ourselves in this way out of concern for simplicity, we are limited to using the risk-free interest rate as our single expected rate of return for the reason given above.
Finally, the reason this method is called "risk-neutral pricing" is that, using the adjusted pricing probabilities, all securities are assumed to have an equal expected rate of return, which is consistent with a risk-neutral market.
Friday, October 26, 2012
Put-Call Parity
The "Put-Call Parity" equation is an instance of the Law of One Price. Its mathematical form is
C - P = S - X * exp(-rT),
where S is the price of a stock at time 0, C is the price at time 0 of a European call option on S with strike price X and expiration time T, P is the price at time 0 of a European put option on S with strike price X and expiration time T, and r is the risk-free interest rate (assumed to be constant).
The term X * exp(-rT)in the equation is constant, and it represents the value at time 0 of the amount X paid with certainty at time T. Figuring out the value of a call and a put with the given strike for any given price of the stock on the expiration date will show that being long such a call and short such a put will always have the save payoff at time T as the stock itself less a payment of X at the same instant, which invokes the Law of One Price.
Put-Call Parity is model independent, meaning that it will hold regardless of under what model prices are determined. Considered differently, any model that violates Put-Call Parity cannot be realistic.
C - P = S - X * exp(-rT),
where S is the price of a stock at time 0, C is the price at time 0 of a European call option on S with strike price X and expiration time T, P is the price at time 0 of a European put option on S with strike price X and expiration time T, and r is the risk-free interest rate (assumed to be constant).
The term X * exp(-rT)in the equation is constant, and it represents the value at time 0 of the amount X paid with certainty at time T. Figuring out the value of a call and a put with the given strike for any given price of the stock on the expiration date will show that being long such a call and short such a put will always have the save payoff at time T as the stock itself less a payment of X at the same instant, which invokes the Law of One Price.
Put-Call Parity is model independent, meaning that it will hold regardless of under what model prices are determined. Considered differently, any model that violates Put-Call Parity cannot be realistic.
Laws of Price: The Law of One Price
I think it's a useful exercise to try clearly to state as many requirements as possible that apply to efficient free market prices. So from time to time I'll devote a post to one or more of these, which I will call "Laws of Price."
The first that I'll look at is a very widely known law with many implementations: the Law of One Price.
Very simply, the Law of One Price states that if two different portfolios of securities have exactly the same payoff at a future date under all possible outcomes, then they must have the same price. Stated differently, there can be only one price for a given pattern of future cash flows, regardless of how the pattern is achieved.
So, a bet that pays $1 if the Steelers beat the Redskins this coming Sunday, plus a bet that pays $1 if the Redskins beat the Steelers in the same game, plus a bet that pays $1 in the event of a tie in the game, should have the same price as $1 paid after the game Sunday with certainty.* If they don't, then a guaranteed profit can be made by selling the more expensive position and buying the less expensive one: simple arbitrage. (The Law of One Price is sometimes referred to with the qualification "no arbitrage," or "arbitrage free.")
* -- These bets will not in fact add up to the value of $1 paid with certainty, but this is because the sports betting market is not efficient: the bookies and the odds-makers arrange to take a percentage of each bet as their compensation, so making all of these bets will cost more than $1 paid with certainty. Also, the "frictional costs" -- costs associated with transaction fees or expenses from acquiring or disposing of positions -- of attempting arbitrage will allow some deviation of prices away from a singular price. But the prices cannot stray too far apart.
The first that I'll look at is a very widely known law with many implementations: the Law of One Price.
Very simply, the Law of One Price states that if two different portfolios of securities have exactly the same payoff at a future date under all possible outcomes, then they must have the same price. Stated differently, there can be only one price for a given pattern of future cash flows, regardless of how the pattern is achieved.
So, a bet that pays $1 if the Steelers beat the Redskins this coming Sunday, plus a bet that pays $1 if the Redskins beat the Steelers in the same game, plus a bet that pays $1 in the event of a tie in the game, should have the same price as $1 paid after the game Sunday with certainty.* If they don't, then a guaranteed profit can be made by selling the more expensive position and buying the less expensive one: simple arbitrage. (The Law of One Price is sometimes referred to with the qualification "no arbitrage," or "arbitrage free.")
* -- These bets will not in fact add up to the value of $1 paid with certainty, but this is because the sports betting market is not efficient: the bookies and the odds-makers arrange to take a percentage of each bet as their compensation, so making all of these bets will cost more than $1 paid with certainty. Also, the "frictional costs" -- costs associated with transaction fees or expenses from acquiring or disposing of positions -- of attempting arbitrage will allow some deviation of prices away from a singular price. But the prices cannot stray too far apart.
Wednesday, September 26, 2012
Understanding Utility Theory
Uncertainty and risk have long confounded human reasoning: the syllogisms of basic logic assume certainty in all assertions, and allow no conclusions to be drawn until all factual states are decided.
And, in practice, people often demonstrate themselves to be remarkably poor at making decisions based on uncertain conditions, foregoing much needed insurances while over-paying for absurdly unlikely coverage, for example.
Within this confusion, the intuitively appealing notion of a "risk premium" has arisen. In general, the notion of a risk premium is that reasonable people will not accept risk without an expectation of, on average, profit above and beyond what is available without risk.
Utility theory is a method of quantifying the notion of a risk premium.
The main premise of utility theory is that we should concern ourselves not with the cash value ("wealth") of anything, but rather with the utility (which I guess is just a of saying "usefulness" with a word derived from Latin ...) of the wealth. And then there are just two parameters that the utility function of wealth are held to have to satisfy: the utility of more wealth is always greater than the utility of less wealth; and the rate of increase of utility relative to wealth decreases as the absolute level of wealth increases.
Mathematically, these two properties are stated as follows:
And, in practice, people often demonstrate themselves to be remarkably poor at making decisions based on uncertain conditions, foregoing much needed insurances while over-paying for absurdly unlikely coverage, for example.
Within this confusion, the intuitively appealing notion of a "risk premium" has arisen. In general, the notion of a risk premium is that reasonable people will not accept risk without an expectation of, on average, profit above and beyond what is available without risk.
Utility theory is a method of quantifying the notion of a risk premium.
The main premise of utility theory is that we should concern ourselves not with the cash value ("wealth") of anything, but rather with the utility (which I guess is just a of saying "usefulness" with a word derived from Latin ...) of the wealth. And then there are just two parameters that the utility function of wealth are held to have to satisfy: the utility of more wealth is always greater than the utility of less wealth; and the rate of increase of utility relative to wealth decreases as the absolute level of wealth increases.
Mathematically, these two properties are stated as follows:
- dU/dW > 0; and
- d²U/dW² < 0.
There are, however, a couple more things to know about utility theory.
The first is that, in standard financial theory today, utility theory is the beginning and end of how risk is understood. This can perhaps be seen most clearly through a 1963 article by Paul A. Samuelson, "Risk and Uncertainty: A Fallacy of Large Numbers", and through a 1995 article that is largely a reprise of the same argument: "On the Risk of Stocks in the Long Run" by Zvi Bodie (sorry, no link).
"Risk and Uncertainty" relies upon utility theory to demonstrate that if you will not accept, due to risk concerns, a single wager with a positive expected return, you also should not accept any multiple repetitions of the same wager, even if the possibility of any loss under the multiple repetitions is almost zero. "On the Risk of Stocks" uses Black-Scholes options pricing theory -- which is based on utility theory -- to dismiss the proposition that, if your time horizon is long enough, the higher expected return on stocks will make them a more suitable investment than investments with more certain returns, even though you would want the more certain investments for a shorter time horizon.
"Risk and Uncertainty" relies upon utility theory to demonstrate that if you will not accept, due to risk concerns, a single wager with a positive expected return, you also should not accept any multiple repetitions of the same wager, even if the possibility of any loss under the multiple repetitions is almost zero. "On the Risk of Stocks" uses Black-Scholes options pricing theory -- which is based on utility theory -- to dismiss the proposition that, if your time horizon is long enough, the higher expected return on stocks will make them a more suitable investment than investments with more certain returns, even though you would want the more certain investments for a shorter time horizon.
The second thing to know about utility theory is that it seems to have almost no predictive value whatever in determining how economic agents, like investors, actually behave. In traditional science, having no predictive value is the hallmark of a hypothesis (not a theory ...) that must be rejected.
Tuesday, September 4, 2012
Introduction
I hope to create with this blog a resource with which people may learn about the theory and practice of options trading, at least as I do it.
I believe that most investors can fine-tune their investment portfolios through the use of options, reducing risk and enhancing return more efficiently than can be done with basic securities. Part of this is due to the essential nature of options, which allow investment return profiles to be shaped to closely fit the perceptions and preferences of the investor. Another part is due to what I believe is a systemic error in how options are priced today.
The entries in this blog will generally be of two types: discussions of theory, and live-tracking of trading in my most active trading account. If anyone happens to read the blog, I'll try to respond to comments either directly or in future blog entries.
I believe that most investors can fine-tune their investment portfolios through the use of options, reducing risk and enhancing return more efficiently than can be done with basic securities. Part of this is due to the essential nature of options, which allow investment return profiles to be shaped to closely fit the perceptions and preferences of the investor. Another part is due to what I believe is a systemic error in how options are priced today.
The entries in this blog will generally be of two types: discussions of theory, and live-tracking of trading in my most active trading account. If anyone happens to read the blog, I'll try to respond to comments either directly or in future blog entries.
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